Intersection theory for o-minimal manifolds

Annals of Pure and Applied Logic 107 (1-3):87-119 (2001)
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Abstract

We develop an intersection theory for definable Cp-manifolds in an o-minimal expansion of a real closed field and we prove the invariance of the intersection numbers under definable Cp-homotopies . In particular we define the intersection number of two definable submanifolds of complementary dimensions, the Brouwer degree and the winding numbers. We illustrate the theory by deriving in the o-minimal context the Brouwer fixed point theorem, the Jordan-Brouwer separation theorem and the invariance of the Lefschetz numbers under definable Cp-homotopies. A. Pillay has shown that any definable group admits an abstract manifold structure. We apply the intersection theory to definable groups after proving an embedding theorem for abstract definably compact Cp-manifolds. In particular using the Lefschetz fixed point theorem we show that the Lefschetz number of the identity map on a definably compact group, which in the classical case coincides with the Euler characteristic, is zero

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References found in this work

One-dimensional groups over an o-minimal structure.Vladimir Razenj - 1991 - Annals of Pure and Applied Logic 53 (3):269-277.
Groups of dimension two and three over o-minimal structures.A. Nesin, A. Pillay & V. Razenj - 1991 - Annals of Pure and Applied Logic 53 (3):279-296.

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